Bring the milk almost to a boil in a medium pot over medium heat. Add the chestnuts and remove the pot from the heat. Cover and let them steep to soften for 30 minutes. Drain the chestnuts, discarding the milk. Chop the chestnuts neither too fine, nor too coarse. Transfer to a medium bowl and add the almonds and chocolate, and gently stir to combine. Set aside. Preheat the oven to 350°. Grease a 10-inch springform pan with a little butter. Line the bottom of the pan with parchment paper, then grease the paper, using a little more butter. Set the pan aside.

Beat the remaining 8 tablespoons of the butter and 1/4 cup of the sugar together in a large mixing bowl with an electric mixer on medium-high speed until light and fluffy. Add the egg yolks one at a time, beating well after each addition. Add the chestnut mixture and stir until well combined. Set aside.

Beat the egg whites in another large mixing bowl with an electric mixer on high speed until frothy. Gradually add the remaining 1/4 cup of sugar, beating until stiff, shiny peaks form. Fold the whites into the batter in thirds. Pour the batter into the prepared pan. Bake until the cake is set and a toothpick inserted into the center comes out clean, 50 60 minutes.

Let the cake cool on a rack before removing it from the pan. To remove it, invert the cake onto a small baking sheet. Remove the pan bottom and parchment paper, then invert the cake onto a cake plate. Dust the cake with powdered sugar just before serving. It’s extra-lovely with dollops of unsweetened whipped cream.